The remonstrance came about largely as a result of the rise of the New Cromwellians. They in themselves were an expression of strong latent support for monarchy and the English traditional constitutional limits on its power, a desire to lose the military overtones of the earlier Protectorate and the increasingly small level of control Cromwell was able to exert due to ill health and frustration with a lack of revolutionary ideology amongst his subjects.
The Humble Petition and Advice wanted to offer hereditary monarchy to Oliver Cromwell, assert Parliament's control over issuing new taxation, provide an independent council to advise the king, safeguard 'Triennial' meetings (every three years) of Parliament and reduce the size of the army in order to save money, amongst other things. These had the effect of limiting, not increasing, Cromwell’s power.
Cromwell refused the Crown, on the 8 May 1657. This may have been because he feared disaffection in the Army, especially considering the proposed reduction in the size of the army, was distressed by allegations of dynastic/personal ambition, did not genuinely accept that a monarchy was necessary in England, or because he feared re-instating the monarchy on the basis that he believed it had been judged by God in the period following the English Civil War.
The Humble Petition was modified to remove the clause on kingship and due to the Naylor case the Humble Petition and Advice was also modified to include a second chamber.
On 25 May, Cromwell ratified a modified Humble Petition and Advice, saying he would nominate his successor as Lord Protector.
Noble, Mark. Memoirs of the protectoral-house of Cromwell: deduced from an early period, and continued down to the present time; ... collected chiefly from original papers and records, ... together with an appendix: ... Embellished with elegant engravings. Volume 1, The third edition, Printed for G. G. J. and J. Robinson, 1787. page 416.