The common law of England, as well as so much of the statute law as  have been heretofore adopted in practice in this state, shall remain in  force, unless they shall be altered by a  future law  of the Legislature;  such parts only excepted as are repugnant to the rights and privileges  contained in this constitution and the declaration of rights, & c. agreed  to by this convention.  Del. Const. Art. 25 (1776)

All  the  laws  of  this  state,  existing  at  the  time  of  making  this  constitution, and not inconsistent with it, shall remain in force, unless  they shall be altered by  future laws; and all actions and prosecutions  now pending, shall proceed as if this constitution had not been made.   Del. Const. Art. VIII, Sec. 10 (1792)

All  the  laws  of  this  State  existing  at  the  time  of  making  this  Constitution and not inconsistent with it, shall remain in force unless  they shall be altered by  future laws; and all actions and prosecutions  now pending shall proceed as if this Constitution had not been made.   Del. Const. Art. VII, Sec. 9 (1831) 

All the laws  of this  State  existing  at the time this Constitution  shall  take effect, and not inconsistent with it shall remain in force, except so  far as they shall be altered by future laws.  Del. Const. Schedules Sec.  18 (1897) 

Degerberg v. McCormick, 184 A. 2d 468 - Del: Court of Chancery 1962 states:
At common law, of course, a person could adopt any name which he chose so long as the change of name was not for an improper purpose. In most states, as in Delaware, there now exist statutes which authorize a proceeding to judicially effect a change of name. Such statutes are universally held not to affect the common law right. They are regarded as merely providing a procedure to establish a court record of the change.

Masjid Muhammad-DCC v. Keve, 479 F. Supp. 1311 - Dist. Court, D. Delaware 1979 states at footnote 13:
The Delaware courts have held that even without pursuing the statutory procedure for a change of name, there exists a common law right to change one's name without court process. See Degerberg v. McCormick, 40 Del.Ch. 471, 184 A.2d 468 (1962). Since the defendants do not dispute that plaintiffs have adopted Muslim names in a manner consistent with the common law, all plaintiffs stand in the same position.