The House's tradition of a ceremonial mace descends from the British House of Commons:
Ceremonial maces originated in the Ancient Near East, where they were used as symbols of rank and authority across the region during the late Stone Age, Bronze Age, and early Iron Age. ....
The earliest ceremonial maces in France and England were practical weapons intended to protect the king's person, borne by the Sergeants-at-Arms, a royal bodyguard established in France by Philip II, and in England probably by Richard I, ....
As the custom of having sergeants' maces began to die out about 1650, the large maces borne before the mayor or bailiffs came into general use. Thomas Maundy functioned as the chief maker of maces during the English Commonwealth. He made the mace for the House of Commons in 1649. This mace is still in use today, though without the original head.
The tradition continues for much the same reason as why the House of Representatives shares the same moniker prefix as the British House of Commons - it addresses the same legislative function. As in the British tradition, the U.S. Constitution specifies that:
Article II Section 7
All bills for raising revenue shall originate in the House of Representatives; ....
Although the U.K. House of Lords now also uses a ceremonial mace that tradition originates only in 1876, as a supplement to the function of the Gentleman Usher of the Black Rod, long after the U.S. Senate established its own procedures.
A search of the House Journal for the first five sessions of Congress for "sergeant-at-arms", "serjeant-at-arms", and "mace" (in relevant contexts) turns up on Monday, April 13, 1789:
The House proceeded to consider the report from the committee appointed to prepare such further rules and orders of proceeding as may be proper to be observed in this House, which lay on the table, and the said report was read, and is as followeth:
"Resolved, That it is the opinion of this committee, that the rules and orders following ought to be established as additional standing rules and orders of this House, to wit:
It shall be the office and duty of a Serjeant-at-Arms, to attend the House during its sitting, to execute the commands of the House, from time to time, and all such process, issued by authority thereof, as shall be directed to him by the Speaker, and either by himself, or special messengers appointed by him, to take and detain in his custody members or other persons ordered by the House to be taken or committed.
A proper symbol of office shall the provided for the Sergeant-at-Arms, of such form and device as the Speaker shall direct, which shall be placed on the Clerk's table during the sitting of the House, but when the House is in committee, shall be placed under the table. The Serjeant-at-Arms shall, moreover, always bear the said symbol when executing the immediate commands of the House, during its sitting, returning the same to the Clerk's table when the service is performed.
and on Tuesday, April 14, 1789:
Mr. Boudinot reported, from the committee to whom was re-committed certain clauses of the report for establishing additional rules and orders of proceeding to be observed in this House ... as followeth:
"Resolved, That it is the opinion of this committee, that the rules and orders following ought to be established, as additional standing rules and orders of this House, to wit:
A Serjeant-at-Arms shall be appointed, ....
A proper symbol of office shall be provided far the Serjeant-at-Arms, of such form and device as the Speaker shall direct, which shall be borne by the Sergeant when in the execution of his office.
The current mace "has been in use in the House since 1841 ... [and] is made of 13 thin ebony rods representing the original states ... bound together by the twining silver bands, which are pinned together and held at the top and bottom of the shaft by repoussé silver bands."